Understanding trigonometric relationships by grounding rules in a coherent conceptual structure
Abstract/Contents
- Abstract
- Mathematical cognition relies on a variety of representations, from algebraic rules to diagrams that convey the meaningful relationships expressed in the algebraic rules. My dissertation shows how a diagram that captures the conceptual structure underlying the domain of trigonometry can make symbolic expressions into more meaningful rules that can be applied and generalized successfully. We presented students with trigonometric identity problems, and examined the representations they used. Students most often reported using the unit circle, a diagram that captures the meaning of trigonometric expressions in an integrated way. Those who reported using the unit circle also tended to have higher accuracy. Other students, who did not report using the unit circle or reported using it less often, were more likely to rely on heuristics and rules that had no connection to the meanings of the trigonometric constructs involved. We created a formal rule-based lesson and another lesson which instead grounded relationships in the unit circle, and we randomly assigned students to each lesson. Students in the grounded lesson condition showed improved accuracy both on problems taught in the lesson and on problems held out for transfer. While the unit circle could be used simply as a standalone procedure to solve problems, some students appeared to apply an understanding of the relationships and symmetries in the unit circle to extend and combine rules. Students reported relying slightly more on a rule or formula when solving taught problems than transfer problems. However, across all problem types, students often reported relying on both the unit circle and a rule or formula on a particular problem. As students gained experience solving problems after the grounded lesson, they tended to reduce their active use of the unit circle, as indicated by their self-reported strategies and by their interactions with an external unit circle tool. Across our studies, the unit circle revealed itself to be a strong, coherent conceptual structure, and it creates a fertile ground for learning and understanding trigonometry through the interplay of visuospatial and rule-based approaches. Students learn how to map parts of a symbolic expression onto meaningful properties of the unit circle, and this grounded understanding facilitates application and generalization of rules in order to solve problems successfully. Helping students from all backgrounds master these grounded rules remains a challenge for teachers and cognitive psychologists.
Description
| Type of resource | text |
|---|---|
| Form | electronic resource; remote; computer; online resource |
| Extent | 1 online resource. |
| Place | California |
| Place | [Stanford, California] |
| Publisher | [Stanford University] |
| Copyright date | 2018; ©2018 |
| Publication date | 2018; 2018 |
| Issuance | monographic |
| Language | English |
Creators/Contributors
| Author | Mickey, Kevin William | |
|---|---|---|
| Degree supervisor | McClelland, James L | |
| Thesis advisor | McClelland, James L | |
| Thesis advisor | Frank, Michael C, (Professor of human biology) | |
| Thesis advisor | Schwartz, Dan | |
| Degree committee member | Frank, Michael C, (Professor of human biology) | |
| Degree committee member | Schwartz, Dan | |
| Associated with | Stanford University, Department of Psychology. |
Subjects
| Genre | Theses |
|---|---|
| Genre | Text |
Bibliographic information
| Statement of responsibility | Kevin W. Mickey. |
|---|---|
| Note | Submitted to the Department of Psychology. |
| Thesis | Thesis Ph.D. Stanford University 2018. |
| Location | electronic resource |
Access conditions
- Copyright
- © 2018 by Kevin William Mickey
- License
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).
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